The Areal Reduction Factor (ARF) : a multifractal analysis

Abstract

The Areal Reduction Factor (ARF) [eta] is a key parameter in the design for hydrologic extremes. For a basin of area A, [eta](A, D, 7) is the ratio between the area-average rainfall intensity over a duration D with return period T and the point rainfall intensity for the same D and T. Besides depending on A, D and possibly T, the ARF is affected by the shape of the basin and by a number of seasonal, climatic and topographic characteristics. Another factor on which ARF depends is the advection velocity, Vad, of the rainfall features. Commonly used formulas and charts for the ARF have been derived by smoothing or curve-fitting empirical ARFs extracted from raingauge network records. Here we derive some properties of the ARF under the assumption that space-time rainfall is exactly or approximately multifractal. We do so for various shapes of the rainfall collecting region and for Vad = 0 and Vad [not equal to] 0. When Vad = 0, a key parameter in the analysis is the ratio Ures = Vres/Ve between the "response velocity" Vres = L/D, where L is the maximum linear dimension of the region, and the "evolution velocity" Ve, = Le/De, where Le and De are the characteristic linear dimension and characteristic duration of organized rainfall features. The effect of Vad [not equal to] 0 depends on the shape of the region. For highly elongated basins, both the direction and magnitude of advection are influential, whereas for regular shaped regions only the magnitude Vad matters. We review ways in which rainfall has been observed to deviate from exact multifractality and models that capture such deviations. We show how the ARF behaves when rainfall is a bounded cascade in space and time.

(cont.) We also investigate the effect of estimating areal rainfall from raingauge network measurements. We find that bounded-cascade deviations from multifractality and sparse spatial sampling distort in similar ways the scaling properties of the ARF. Finally we show how one can reproduce various features of empirical ARF charts by using multifractal and bounded cascade models and considering the effects of sparse spatial sampling.